We can employ dimensional extension of quasiperiodic models in \(D>1\)  with spatial modulations along orthogonal axes to obtain quantum Hall effects in \(2\times D>2\) , which are then characterized by higher dimensional Chern numbers, families of associated bulk responses, as well as, associated boundary effects with co-dimension \(d\in \lbrace 1,\ldots ,D\rbrace \)  [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}. For such cases, similar approximant paths apply with generalized translation operators as discussed above. At the same time, it becomes much harder to identify gaps that remain open, and the gap labeling involves bookkeeping that tracks the labels from the orthogonal axes. Furthermore, introducing terms that mix the orthogonal spatial axes, as is expected for more general quasicrystals [cf. Fig. REF (a)], implies that a mapping to a \(2\times D\)  ancestor model may be possible [see, e.g., recent work with hexagonal moiré structures [10]}], but the gap labeling approach detailed here will no longer hold.
